Calculus Examples

$\frac{x^{2}}{\sqrt{x}}$

Step 1

Write $\frac{x^{2}}{\sqrt{x}}$ as a function.

$f (x) = \frac{x^{2}}{\sqrt{x}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{x^{2}}{\sqrt{x}} d x$

Step 4

Simplify the expression.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int \frac{x^{2}}{x^{\frac{1}{2}}} d x$

Step 4.2

Simplify.

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Step 4.2.1

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\int x^{2} x^{- \frac{1}{2}} d x$

Step 4.2.2

Multiply $x^{2}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.2.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{2 - \frac{1}{2}} d x$

Step 4.2.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int x^{2 \cdot \frac{2}{2} - \frac{1}{2}} d x$

Step 4.2.2.3

Combine $2$ and $\frac{2}{2}$ .

$\int x^{\frac{2 \cdot 2}{2} - \frac{1}{2}} d x$

Step 4.2.2.4

Combine the numerators over the common denominator.

$\int x^{\frac{2 \cdot 2 - 1}{2}} d x$

Step 4.2.2.5

Simplify the numerator.

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Step 4.2.2.5.1

Multiply $2$ by $2$ .

$\int x^{\frac{4 - 1}{2}} d x$

Step 4.2.2.5.2

Subtract $1$ from $4$ .

$\int x^{\frac{3}{2}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C$

Step 6

The answer is the antiderivative of the function $f (x) = \frac{x^{2}}{\sqrt{x}}$ .

$F (x) =$ $\frac{2}{5} x^{\frac{5}{2}} + C$